It is evident that in the 1800's after Newton had established his philosophical apprehension of Nature, that British and American philosophers made great headway under the assumption that there were laws whether of Nature, or Natures god, or mind, that logic could reveal to man. And it was believed that a sufficient logic would reveal the orderliness of all things material and spiritual, if these be accepted as valid distinctions .
In old europe, the similar vein of thinking took place under many heads, whether they be French, German or Italian national heroic thinkers and philosophers, or religious clerics. Certainly Russia was not excluded, and despite nationalistic ambitions and the political turmoil and wars, scholars yet debated internationally within the borders of old empirical institutions, scholarly societies, and philosophical and religious conclaves. This loose network of connections, called by many the illuminati, at once mysterious and perfectly clear, fostered the human need as it seems to progress knowledge and insight of nature and natures gods, for whatever principal purpose.
Thus the Knights Templars, the Rosae Crucians, the Masons, regardless of the workings of any secretive inner councils served in the main as societies for the advancement of knowledge. And where no such recognised groupings may be found, yet some guild or society or other collection, secretive or open may exist for the purpose of association and establishment of cherished values.
It is from such associations that public services and schools have arisen, if not from that other grouping the religious society.This being said in explanation of the "other " connection humand have beyond those of church and state which suffice as a means of knowledge promulgation.
Thus what one finds on one continent one also finds on every other in time, but particularly in the 1800's the race was on for scientific and technological prowess as well as superiority in all spheres. That such philosophies and Thinkings should arise then is of historical manifestation, and striking enough for Karl Marx to propose a formative philosophy of History for peoples individuals and ideas,but that need not concern us as yet. It is evident that on all continents many were moving toward similar ideas and descriptions of reality at various paces, and with more or less understanding of the ancient and deep philosophies driving this movement.
For thousands of years PYTHAGOREAN philosophy has underpinned the largely western and islamic, middle eastern nations, buried deep within the religious systems of those countries. That the cultural appearance is so different in each country is testament to the fundamental nature of the philosophy. Where it betrays itself is in architecture and in modes of discourse and in goals of life.
We may search many of the African nations and find pythagorean ideals as impositions upon a much older set of cultural and tribal values, and as such it may be recognised as "westernisation". There are very few peoples who have not been touched by pythagorean idealism in one of its many forms. However, those closest to the source, having access to redactions of Pythagoras' own works have tended to exemplify most the pythagorean school of thought, redacted by Plato. Nevertheless, in their eagerness to divide the spoils academia dissociated itself from the platonic Academy ideals (not willingly) and embraced modernity, thus severing any knowledge connection to the Euclidean working out of platonic pythagoreanism.
Further more, in their headlong rush for the prestige, and thinking themselves greater than they ought, they misconstrued the Euclidean teaching material, and then proceeded to knock down their own misconceptions, claiming that Euclid was rhus and such and usually in some small pecuniary instance "wrong!", in the meanwhile abrogating to themselves what might be "right" in Euclid, and purveying their further "corrections" as new insight, deeper knowledge and the like.However, their protestations have not passed the test of time nearly as well as Euclid's material has, and indeed time has shown these new teachers to be in "error", or more kindly to have revealed the correct intention of Euclid's construction as he designated it.
Translation has come on apace and much which was obscure is now revealed by new evidence to be plainly this or that. What is plain is that Euclid is never so much to be thought of as Elementary or easy or at all simple, but as foundational and fundamental and philosophical, and that in the great treatise that bears his name he provides a course in the theory of space suitable to further and university level education. In fact, in several more books he advances the material to beyond Doctorate level,and on to the level of a professor of such things.
Now it is usual to attack Euclid at the level of his postulates for planes (epiphaneia) in ignorance of the fact that he provides material at the level of Spherical and solid geometry which is of the exact nature of so called non Euclidean geometry! No one is claiming that Euclid had answered it all, but that he had sufficiently laid out a material for his students to continue the study beyond his current knowledge, which indeed Appolonius and Archimedes and others have done.
Where Euclid also has been attacked is in his notation, because, not understanding the pythagorean underpinnings of his material, students never took to the pythagorean extensions of notation symbology, the deeper, often called esoteric, gematrial forms of their workings with notation. So now when logicians and symbolic algebraists purport to lay down the laws of symbols, of their own discovery, they do so anew, ignorant of the formalisms before worked out by the pythagoreans. Indeed many symbols employed by the pythagoreans were assumed and given new referent, thus trampling on old formulae and rituals by which "magical" results were obtained in divination and calculation, by analogical, symbological reasonings, or rather proportionings.
It is exciting to see these ideas re framed in modern language and with modern conceptions underpinning, bur horrendous to see the obscurantism due to ancient language terms: thus latin and greek scholars may have a clue what is being referred to, but the non classically trained reader will hopelessly struggle. This is the more ironic because what is usually referred to is so common in everyday experience that ready examples and synonyms should be readily available. In fact i have often remarked that almos every so called advanced mathematical notion or surface can be found in most modern kitchens and kitchen appliances, including the kitchen sink! Many quantum theoretic notions are to be found in a bowl of bubbly washing up solution for example, and the tap that pours water into that surface!
De Morgan's philosophy therefore, occurring at the same time as Hamilton's and Grssmann's serves to inform on the common conceptions and misconceptions, and to explain how Grassmnn just happens to have had hat peculiar set of circumstances which made him as Leibniz heir, nor Wallis -Newtons, have the sufficiently alternative conceptions to unconsciously tease out the pythagorean principles underpinning Euclidean teaching material,by focusing on an original interpretation of notation symbology. In fact others were driven by a long practice to ignore notation symbology beyond a certain level. Grassmann, while not going into the extremes of pythogorean symbologyat least went up to a sufficient level to perform "magic" with regard to a calculus of space.
Why Leibniz? Leibniz was a Pythagorean who produced a monadic philosophy, which due to historical circumstances was supersceded by technological advances that did not require that level of thinking, and yet threw up new questions which Leibniz could not address due to his death before publication. The brilliance of Leibniz is well worth a read, but few have done so, including myself beyond a scan. It was the rise of Kantian, pro Newtonian philosophy that placed Leibniz on the backburner, despite his far thinking approach to technology, mathematics,philosophy and science.
The industrial revolutin was the Zeitgeist of the time: Pragmatism ruled as king. Grassmnn was thus looking for a pragmatic philosophy that would advance his nation technologically, not merely philosophically. Thus he did not want to dodge the hard questions, the situations where conceptions of logic and philosophy did not easily conform with experience. He did not seek to make reality conform with some ideal, and thus to exclude what did not fit; he sought to learn the laws of mind that mde it fit together at some level. In that regard, the subjective apprehension through symbolic terminology was key,and crucial, and in his Ausdehnungslehre he spends a major and important part driving that message home and showing how to achieve the correct symbolic terminology. Thus he payed close attention to the subjective processing of spatial interaction as foundational to a new "science" of space calculus,seen as a new branch or rather Root of mathematics, but only in the sense of the prevailing belief that space or nature could only be apprehended via mathematics.
Any shortcomings in the Ausdehnungslehre are from his conviction that most other things in mathematics were right: for example the real numbers as a continuum, the imaginary magnitudes as numbers etc to do with number theory. He viewed what he was demonstrating as an extension of the normal mathematics. The term Ausdehnung has a technical meaning in mathematics: it means a formal or terminological addition that extends the applicability of an established method or formula. Thus Hamilton sought to extend the methods of planar couples to three dimensions, by addition of some term or terms that enabled that. De Morgan sought to extend his theory on trigonometry of doubles to trigonometry of triples. Warren sought to extend his notions on representing the imaginaries in the plane through doubles to triples, the very act of collaboration with him in this lead to Hamilton hitting upon the Quaternions.
Thus Ausdehnungslehere is not about vectors or lineal Algebra, but about how to extend any system to a more general system!Grassmann did not purport to deal with "measurement " in a piecemeal stepwise fashion, and this creeping way to attain gradually to greater and greater generality; he purports to lay down the rules and laws of extension themselves. Thus he r=proposes to lay out a theory of how any extension to any level may proceed: that in any subject one may pprehend that subject in any spatial form as desired without hindrance. Thus taking Hamilton's quest to determine the laws of the imaginaries ,as in th plane, and then to determine it for the triples and so on, Grassmann purports to give it all at once for any number of "dimensions".
Such an idea and undertaking therefore could not be rushed or attained to easily, nor with any certainty of success. It is therefore testament to the fat that this notion only gradually dawned on Grassmann, because any one would have been surely daunted and hopes dashed at the sheer boldness of the conception. It is only after having considerable personal success and insight in employing his discovries that he ventured to publish the audacious plan, hoping that what he had achieved so far would garner support and fellow workers in the field, and thus together they would push on in research toward the eventual goal: a complete and rigorous theory of how to extend to any number of dimensions any formulation of spatial relationships.
Fortunately nobody could understand his personal communication style, although i may add that it is in most parts quite clear. It is his description of subjective experiences that he is having which are most confusing. This wasdue to the fact that he wrote under pressure of work and had little time to proof read and check for clarity.
After 15 years of disappointing response it dawned on him hat he had a communication difficulty. He repeats the dictum: the course of study must be suited to the expected Audience! He chose to write to a philosophical audience but claimed the content was "mathematical": He then rewrote the material for a mathematical audience published in 1862, only to find to his horror that the material was bereft of all resonance with his previous work, although the same results were achieved. Consequently he rewrote it for publication with copious references to the ideas, sentiments and understandings in his firs book, even including quotes from his first book as well as foot notes. However, this book ,written for mathematicians he boldy proclaims as a Science! So now scientists may be attracted but mathematicians put off!
Fortunately a growing number of mathematical scientists came to learn about the material before Grassmann died. However, it was only a few who understood what Grassmann was struggling to do. AN Whitehead was one, who attempted with Bertrand Russel to incorporate Grassmann's ideas of a theory of how to extend to any dimensions, with his own rigorous research into the logical and symbolic foundations of a "subject" called mathematics. Unfortunately both he and Russel formally attempted to derive the number system , a crucial element of Grassmann's approach which he just assumed as correct, and found that many symbolic and predicate difficulties lay at the heart of that endeavour. Thus from a fundamental point of view Russel and Whitehead showed that any system that relies on a symbolic, set definition of number is "flawed" by predicate paradox.
Whitehead never completed his mammoth undertaking, nor did Russel. I have to assume that they eventually became disheartened by as grassmann once put it " the dark ways of abstraction!" Grassmann of course was made of sterner stuff.
The problem with number, at the foundation of "mathematics" is the problem with human misconception. For Grassmann there was no "mathematics" just space. He had been told that the language of mathematics was the one to understand space, but almost as soon as he started his steps to Ausdehnunslehre he realised that many subjects and fields of study had arbitrary boundaries drawn, including mathematics, because of which a common and agreed terminology would be necessary to proceed with his plan successfully, that is to avoid confusion. Thus he would have identified the number and predicate problems and moved to some better all encompassing notion like proportion and adjectival quantifiers,or intensifiers.
Grassmann's work, i think lives on, but not in vector theory.It lives on in computer Science in which the issue of extending a set of space referents to cover any dimension of reference compactly and fractally is alive and kicking.